Quantal time asymmetry: mathematical foundation and physical interpretation

For a quantum theory that includes exponentially decaying states and Breit-Wigner resonances, which are related to each other by the lifetime-width relation $\tau =\frac{\mathrm{\hslash }}{\mathrm{\Gamma }}$, where $\tau$ is the lifetime of the decaying state and $\mathrm{\Gamma }$ the width of the resonance, one has to go beyond the Hilbert space and beyond the Schwartz-Rigged Hilbert Space $\mathrm{\Phi }\subset \mathcal{H}\subset {\mathrm{\Phi }}^{\times }$ of the Dirac formalism. One has to distinguish between prepared states, using a space ${\mathrm{\Phi }}_{-}\subset \text{mat}hcalH$, and detected observables, using a space ${\mathrm{\Phi }}_{+}\subset \mathcal{H}$, where $-(+)$ refers to analyticity of the energy wave function in the lower (upper) complex energy semiplane. This differentiation is also justified by causality: A state needs to be prepared first, before an observable can be measured in it. The axiom that will lead to the lifetime-width relation is that ${\mathrm{\Phi }}_{+}$ and ${\mathrm{\Phi }}_{-}$ are Hardy spaces of the upper and lower semiplane, respectively. Applying this axiom to the relativistic case for the variable $\text{s}=p_{\mu }{p}^{\mu }$ leads to semigroup transformations into the forward light cone (Einstein causality) and a precise definition of resonance mass and width.